If z^3 =x^2 −y^2 , → { (((dx/dt)=3)),(((dy/dt)=2)) :} find (dz/dt) when x=4 and y=1


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Question Number 125996 by bramlexs22 last updated on 16/Dec/20
$If z^3 =x^2 −y^2 , → { (((dx/dt)=3)),(((dy/dt)=2)) :} find (dz/dt) when x=4 and y=1$
$I f z^{3} = x^{2} - y^{2}, \to {\begin{cases} \frac{d x}{d t} = 3 \\ \frac{d y}{d t} = 2 \end{cases}$ $f i n d \frac{d z}{d t} w h e n x = 4 a n d y = 1$
	Answered by Olaf last updated on 16/Dec/20

	$z^{3} = x^{2} - y^{2}$ $\Rightarrow 3 z^{2} \frac{d z}{d t} = 2 x \frac{d x}{d t} - 2 y \frac{d y}{d t} (1)$ $x = 4, y = 1 \Rightarrow z^{3} = 4^{2} - 1^{2} = 1 \bar{5}, z = 15^{\frac{1}{3}}$ $(1) : 3 . 15^{\frac{2}{3}} \frac{d z}{d t} = 2 \times 4 \times 3 - 2 \times 1 \times 2 = 20$ $\frac{d z}{d t} = \frac{20}{3 . 15^{\frac{2}{3}}} = 3^{- \frac{5}{3}} . 4 . 5^{\frac{1}{3}}$
	Commented by bramlexs22 last updated on 16/Dec/20

	$t h a n k s$
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